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Heat Transfer Search Algorithm for Sizing Optimization of Truss Structures Abstract Heat transfer search (HTS) is a novel metaheuristic optimization algorithm that simulates the laws of thermodynamics and heat transfer. In this study, the HTS algorithm is adapted to truss structure optimization. Sizing optimization searches for the minimum weight of a structure subject to stress and displacement constraints. Three truss structures often taken as benchmarks in the optimization literature are selected here in order to verify the efficiency and robustness of the HTS algorithm. Optimization results indicate that HTS can obtain better designs (i.e. lighter trusses) than most of the state-of-the-art metaheuristic optimizers. The convergence behaviour of HTS also is as good as the other algorithms. Keywords Heat transfer search; metaheuristic search algorithms; sizing optimization; truss structures.
S.O. Degertekin a L. Lamberti b M.S. Hayalioglu a a
Civil Engineering Department, Dicle University, 21280,Diyarbakir, Turkey,
[email protected],
[email protected] b Matematica e Management, Politecnico di Bari, 70126, Bari, Italy,
[email protected] http://dx.doi.org/10.1590/1679-78253297 Received 11.08.2016 Accepted 07.12.2016 Available online 09.12.2016
1 INTRODUCTION The main philosophy of metaheuristic optimization algorithms is to perform a pseudo-random search mimicking some natural phenomenon. Among methods developed in the last two decades, particle swarm optimization (PSO) reproduces the social behaviour of swarms (Kennedy and Eberhart, 1995); harmony search (HS) simulates the natural performance processes of musicians searching for a state of harmony (Geem et. al, 2001); artificial bee colony (ABC) is another swarm intelligence method which mimics the intelligent behaviour of honey bee swarms (Karaboga, 2005); big bang-big crunch (BB-BC) reproduces the process of expansion-contraction of the universe (Erol and Eksin, 2006); charged system search (CSS), developed by Kaveh and Talatahari (2010), utilizes the Newtonian law of mechanics in addition to the electrical physics laws to direct the agents in order to recognize the optimum locations; firefly algorithm (FFA) is inspired by social behaviour of fireflies and the phenomenon of bioluminescent communication (Yang, 2010); teaching-learning-based
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optimization (TLBO), developed by Rao et al. (2011), mimics the teaching and learning processes in a classroom, in particular influence of a teacher on learners and the mutual interactions of learners; flower pollination algorithm (FPA) which simulates the pollination process of flowering plants, firstly proposed by Yang (2012); swallow swarm optimization (SSO), developed by Neshat et al. (2013), bases on the analogy between swallow swarm behaviours and optimization problems; water evaporation optimization (WEO), developed by Kaveh and Bakhshpoori (2016), mimics the evaporation of a tiny amount of water molecules adhered on a solid surface with different wettability which can be studied by molecular dynamics simulations. Various metaheuristic optimization methods have been applied to sizing optimization of truss structures (see, for example, the reviews by Lamberti and Pappalettere (2011), Saka and Dogan (2012), and the textbook by Kaveh (2014)). Just to mention a few examples, Sonmez (2011) proposed an artificial bee colony (ABC) algorithm with an adaptive penalty function approach (ABCAP) to minimize weight of truss structures; ABC-AP algorithm was found to be competitive in terms of optimized weight but showed very poor convergence capability compared with other metaheuristic algorithms. Degertekin (2012) developed improved harmony search algorithms called efficient harmony search (EHS) and self-adaptive harmony search (SAHS) for sizing optimization of truss structures. The robustness of the proposed methods was verified by solving four design examples. The results demonstrated that SAHS is very powerful compared to classical harmony search and other metaheuristic optimization methods. Teaching-learning based optimization (TLBO) was used for optimum design of truss structures by Degertekin and Hayalioglu (2013); the efficiency of the proposed implementation was verified in several truss design examples. Camp and Farshchin (2014) proposed a modified teaching–learning-based optimization (TLBO) algorithm for optimization of truss structures. Without considering population size, convergence criterion, and penalty function structure, TLBO is parameter insensitive. The performance of above mentioned modified TLBO was found to be equivalent to other metaheuristic methods without applying parameter-based search mechanisms. Firefly algorithm (FFA) was applied to optimum design of truss structures by Degertekin and Lamberti (2013). FFA proved itself to be very competitive with other metaheuristic optimization methods. Kaveh et al (2014) hybridized the particle swarm and swallow swarm optimization (HPSSO) to solve mathematical optimization problems and truss weight minimization problems. The results obtained from design examples prove that HPSSO outperforms other PSO variants and is very competitive with state-of-art metaheuristic methods. Bekdaş et al. (2015) developed an algorithm called flower pollination algorithm (FPA) for sizing optimization of truss structures. The design examples presented in their study showed that FPA could produce better results in some cases. However, it was concluded that detailed parametric study should be performed in order to find the best parameter setting for the FPA algorithm so that it can solve a wider group of structural optimization problems. Kaveh and Bakhshpoori (2016) tested the water evaporation optimization (WEO) algorithm in six truss design problems from small to normal scale and compared it with the most effective avail-
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able state-of-the-art metaheuristic optimization methods. WEO resulted very competitive in terms of solution quality and robustness and the only weak point of the algorithm is its low convergence speed. A novel metaheuristic search method called heat transfer search (HTS) has been recently developed by Patel and Savsani (2015) for solving optimization problems. HTS simulates the course of action followed by a system to reach thermal equilibrium. The efficiency of HTS was evaluated through 24 mathematical optimization problems with explicit cost function and constraints. A detailed comparison with a variety of metaheuristic methods (besides PSO, ABC and TLBO, also genetic algorithms, differential evolution and biogeography-based optimization were considered) was carried out setting for all algorithms the same limit number of function evaluations or the same convergence tolerance with respect to the target optimum. Numerical results demonstrate the efficiency of HTS compared to other metaheuristic methods. The main goal of this study is to introduce the HTS algorithm into the optimization of truss structures. The performance of HTS is evaluated by considering three truss structures with 25, 72 and 200 elements. For that purpose, HTS is compared with recently developed metaheuristic optimization methods such as artificial bee colony algorithm with adaptive penalty (ABC-AP), selfadaptive harmony search algorithm (SAHS), teaching-learning based optimization (TLBO), firefly algorithm (FFA), hybrid particle swarm swallow swarm optimization (HPSSO), flower pollination algorithm (FPA) and water evaporation optimization (WEO). The considerable amount of data available in the literature for the selected test problems provides a valuable basis of comparison to evaluate the performance of HTS. The remainder of this paper is organized as follows: the structural optimization problem is stated in Section 2. The HTS algorithm is explained in Section 3. The implementation of HTS algorithm for optimization of truss structures is presented in Section 4. The design examples are described in Section 5. Finally, some concluding remarks are presented in Section 6.
2 THE STRUCTURAL OPTIMIZATION PROBLEM The sizing optimization problem of a truss structure including nm members can be formulated as: Find
X [ x1 , x 2 ,....., x ng ] ,
to minimize
ng
mk
i 1
k 1
ximin xi ximax
i=1,2,….,ng
W ( X ) xi k Lk
(1) (2)
subject to
mc m mt ,
m=1,2,….,nm
(3)
d j ,min d j d j ,max ,
j=1,2,…..,ndof
(4)
where: X is the vector containing the design variables; xi is the cross-sectional area of the i-th group of bars, taken as the i-th design variable; ximin and ximax , respectively, are the minimum and maximum values for cross-sectional areas; W(X) is the weight of the structure; ng is the number of
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design variables, equal to the number of member groups included in the structure; mk is the totalnumber of members in group k ; ρk and Lk, respectively, are the mass density and the length of the k-th member in the i-th group. m is the axial stress of the m-th member; mc and mt , respectively, are the allowable compression and tension stresses for the m-th member. d j is the nodal displacement of the j-th translational degree of freedom, d j ,min
and d j ,max , respectively, are its lower
and upper limits; ndof is the number of translational degrees of freedom. The design constraints given in Eqs. (3-4) are handled by using the following modified feasiblebased mechanism, successfully applied to sizing optimization of truss structures (Kaveh and Talatahari, 2009a; Degertekin and Hayalioglu, 2013): (i) Any feasible design is preferred to any infeasible design; (ii) Infeasible designs with a slight constraint violation are taken feasible; (iii) Between two feasible designs, the one having the better objective function value is preferred; (iv) Between two infeasible designs, the one having the smaller constraint violation is preferred.
3 THE HEAT TRANSFER SEARCH ALGORITHM Heat transfer is the branch of Physics concerned with the exchange of heat between systems having different temperatures. Temperature gradients result in a transport of thermal energy within a system or between systems in thermal contact to each other. Heat transfer occurs because any system attempts to reach the temperature of its surroundings (Hollman, 2010; Çengel, 2008; von Böckh and Wetzel, 2012). Clusters of molecules possess different temperature levels in heat transfer. If a system is thermally unbalanced with itself and/or its neighbouring systems, it attempts to overcome this situation by reaching a state of thermal equilibrium. Heat transfer consists of three basic mechanisms: conduction, convection and radiation. Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. Convection is a mode of heat transfer between a solid surface and the adjacent liquid or gas that is in motion; it involves the combined effects of conduction and fluid motion. Radiation is the energy emitted by the matter in the form of electromagnetic waves as a result of the changes in the electronic configurations of the atoms or molecules (Hollman, 2010; Çengel, 2008; von Böckh and Wetzel, 2012). Heat transfer has many application areas such as heating, ventilating and air conditioning systems, thermal power plants, refrigerators and heat pumps, gas separation and liquefaction, cooling of machines, processes requiring cooling or heating, heating up or cooling down of production parts, rectification and distillation plants etc. The detailed information about the thermal equilibrium and heat transfer can be found in the study of Patel and Savsani (2015) and other sources (Hollman, 2010; Çengel, 2008; von Böckh and Wetzel, 2012). Therefore, the same definitions and equations will not be repeated here for the sake of brevity. The laws of thermodynamics and heat transfer have been incorporated by Patel and Savsani (2015) into the Heat Transfer Search (HTS) metaheuristic optimization algorithm, developed for constrained optimization problems. The HTS algorithm consists of three phases called as ‘conduction phase’, ‘radiation phase’ and ‘convection phase’. The ‘conduction phase’, ‘radiation phase’ and ‘convection phase’ neutralize the thermal unbalance (i.e. change the energy level) of the system by conduction, radiation and convection heat transfer, respectively. A uniformly distributed random Latin American Journal of Solids and Structures 14 (2017) 373-397
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number (Rn) between 0 and 1 is initially generated in the HTS algorithm in order to decide which phase should be executed. One of these phases is used in an iteration according to the value of Rn and each phase has equal probability between 0 and 1 to be carried out. It is demonstrated that 0– 0.3333, 0.3333–0.6666 and 0.6666–1 are suitable intervals for the values of Rn to execute conduction, radiation and convection phases, respectively. Initial population is generated randomly in the HTS algorithm similar to other population-based optimization algorithms. After that, new designs are produced by using the conduction, convection or radiation phases. If a new trial design yields a better objective function value than the existing one, the previous design is replaced. Otherwise, the original design is left unchanged. Moreover, worst designs of the current iteration are replaced with elite designs of the previous iteration if the elite designs of the previous iteration have better objective function values than the current worst ones. Another rule applied in the HTS algorithm is that if duplicate designs are found in the population after replacing worst designs with elite designs, one of the duplicate designs is modified. For this purpose, a randomly selected design variable of the duplicate design is updated as follows: if
0.0 ri 0.50
(5)
old x new x old if j j (1 ri ) x j
0.50 ri 1.0
(6)
x new x old ri x old j j j
old where: x new j and x j , respectively, are the new and old values of the selected j-th design variable; ri
is a random number generated between 0 and 1. The analogy between the HTS and optimization of truss structures can be established as follows: different temperatures of molecules represent the different design variables (member groups for a truss design), the energy level of the molecules symbolizes the objective function of the truss structure, the cluster of molecules in the heat transfer represents candidate designs in the population of HTS algorithm. The current best truss design is taken as the surroundings and rest of the truss designs are considered as a system. 3.1 Conduction Phase The conduction phase is executed if the uniformly distributed random number (Rn) generated is between 0 and 0.3333. In this phase, designs are modified based on the randomly selected design from the population and only one randomly selected design variable is updated. This phase includes two parts. According to the iteration number (it) and the conduction factor (CDF), the first part is executed as follows. If it
